The lower limit topology on has basis of a topology . Every such nonempty interval contains a rational, so this topological space is separable. Its topology is finer than the usual Euclidean topology; the one-sided basic neighbourhoods allow its square to have an uncountable discrete antidiagonal.
The Sorgenfrey plane is the product topology on two copies of the Sorgenfrey line. It is separable because meets every nonempty basic rectangle . Its antidiagonal is an uncountable discrete subspace: meets it only at . In a discrete space every dense subset is the whole space, so this subspace is not separable. Thus separability need not pass to subspaces of nonmetrizable spaces.
// Target: analysis.bigb
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Lower limit topology, also known as the standard topology on the real numbers, is a specific topology defined on the set of real numbers \(\mathbb{R}\). This topology is generated by a basis consisting of all half-open intervals of the form \([a, b)\) where \(a < b\).